I seriously doubt that Edward can read animal's minds. It's not like they speak a language he knows. It would be really weird if Edward learned how to speak mountain lion, wouldn't it?
Another reason for the doubt is that, don't you think he would have mentioned it when Bella was hunting in Breaking Dawn? I mean, that would certainly be a conversation starter. "So, love, did you know that that elk you just devoured was thinking that....." I am pretty positive if that was possible he would have told her.